In studying terminal constraints, compact control sets are not satisfactory since we would not be able to show, in full generality, that the constraint can indeed be reached with a finite cost. Part of the challenge of the paper is to find a framework in which the process is sufficiently "controllable" but the HJB equation is still solvable. For that we need to impose restrictions on the coefficients. In particular, we require the agent to control separately the drift and the volatility. We also require some growth assumptions on the Hamiltonians. This allows us to use the weak Bernstein method as in Ishii and Lions [1]}, Barles [2]}, Lions and Souganidis [3]} and Armstrong and Cardaliaguet [4]} (among others) to prove some Lipschitz estimates on the (viscosity) solution of the HJB equation. These estimates allow us to apply the regularity results for compact control sets as in [5]}.
